Non-chiral Intermediate Long Wave equation and inter-edge effects in narrow quantum Hall systems
aa r X i v : . [ m a t h - ph ] O c t Nonchiral Intermediate long-wave equation and interedge effects in narrow quantumHall systems
Bjorn K. Berntson, ∗ Edwin Langmann, † and Jonatan Lenells ‡ Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden Department of Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden (Dated: October 27, 2020)We present a nonchiral version of the Intermediate long-wave (ILW) equation that can modelnonlinear waves propagating on two opposite edges of a quantum Hall system, taking into accountinteredge interactions. We obtain exact soliton solutions governed by the hyperbolic Calogero-Moser-Sutherland (CMS) model, and we give a Lax pair, a Hirota form, and conservation laws forthis new equation. We also present a periodic nonchiral ILW equation, together with its solitonsolutions governed by the elliptic CMS model.
I. INTRODUCTION
One important feature of the Fractional QuantumHall Effect (FQHE) is the strikingly high accuracy bywhich the Hall conductance, σ H , is measured in unitsof the inverse von Klitzing constant, e /h . Therefore,satisfactory explanations of these FQHE measurements, σ H h/e = , , , . . . , must be based on exact analyticarguments, and theories of the FQHE have close con-nections to integrable systems. Two important classesof integrable systems which are seemingly very differ-ent but which are both connected with the FQHE are(i) Calogero-Moser-Sutherland (CMS) models describ-ing FQHE edge states, and (ii) soliton equationsof Benjamin-Ono (BO) type describing the dynamicsof nonlinear waves propagating along FQHE edges (background on the soliton equations appearing in thispaper can be found in Section VI B). These systemsare related by a fundamental correspondence betweenCMS systems and BO-type soliton equations, which pro-vides the basis for a mathematically precise derivationof hydrodynamic descriptions of CMS systems. Itis worth noting that this subject has recently receivedconsiderable attention in the context of nonequilibriumphysics.
While the CMS-BO correspondence has been success-fully used to understand FQHE physics, it is incomplete.Indeed, CMS systems come in four types: (I) rational,(II) trigonometric, (III) hyperbolic, and (IV) elliptic, and while the soliton equations related to the rationaland trigonometric cases are well-understood since a longtime, soliton equations related to the hyperbolic andelliptic cases were only recently identified as the Inter-mediate long-wave (ILW) equation and the periodic ILWequation, respectively.
However, as we will show inthis paper, the latter two soliton equation are not unique:there are other equations which are more interesting inthat they are of a different kind and describe new physics.The correspondence between CMS and BO systems ex-ists both at the classical and at the quantum level, and we consider both. As will be explained, we dis-covered the quantum elliptic version of the soliton equa-tion presented in this paper from a second quantization of the quantum elliptic CMS model.
However, the ex-act results on the solution of this equation presented inthis paper are restricted to the classical case for simplic-ity. We first give and prove our results in the hyperboliccase; the generalization to the elliptic case is surprisinglyeasy, as will be shown later on.
Plan.
In Section II A, we give a heuristic argumentmotivating a generalization of the BO equation that de-scribes coupled nonlinear waves propagating in oppo-site directions, and we present this so-called nonchiralILW equation in Section II B. Our quantum results canbe found in Section III: First, the relation between thequantum elliptic CMS model and a quantum version ofthe nonchiral ILW equation is presented (Section III A);second, a detailed motivation of our proposal that thequantum nonchiral ILW equation can describe nonlinearwaves propagating on two opposite edges of a FQHE sys-tem boundary and taking into account interactions be-tween different edges is given, including a review of therelevant background (Section III B). Results that provethat the classical nonchiral ILW equations are exactlysolvable can be found in Sections IV (hyperbolic case)and V (elliptic case). In Section VI, we shortly recall theapplication of the BO- and ILW equations to nonlinearwater waves, and we present a simple physical argumentsuggesting that the nonchiral ILW equation is also rel-evant in that context. We conclude with final remarksin Section VII. Appendix A provides mathematical de-tails, and Appendix B shortly explains numerical compu-tations we performed to test our exact analytic solutions.
II. CLASSICAL PHYSICS DESCRIPTION
As explained in the Section III, we discovered the quan-tum version of the nonchiral ILW in the context of theFQHE. However, for simplicity, we first present in thissection a simpler heuristic argument on the classical levelwhich leads to the classical version of this equation. Aselaborated in Section VI in one example, this heuristicargument can be straightforwardly adapted to other sit-uations, suggesting that the nonchiral ILW will also findother applications in physics.
A. Heuristic motivation
The CMS models can be defined by Newton’s equations¨ z j = − N X k = j V ′ ( z j − z k ) ( j = 1 , . . . , N ) (1)where the two-body interaction potential is V ( r ) = r − in the rational case and V ( r ) = ( π/L ) sin − ( πr/L ), L >
0, in the trigonometric case (the arguments inthis paragraph apply to both cases). Eq. (1) describesan arbitrary number, N , of interacting particles with po-sitions z j ≡ z j ( t ) at time t . While one often restrictsto real positions when interpreting the CMS model as adynamical system, one has to allow for complex z j whenstudying the relation to the BO equation; this gener-alization preserves the integrability. The CMS model isinvariant under the parity transformation P : z j → − z j for all j . However, the corresponding BO equation is notparity-invariant: it is given by u t + 2 uu x + Hu xx = 0,where u ≡ u ( x, t ) and H is the Hilbert transform (inthe rational case, ( Hf )( x ) = (1 /π ) − R R ( x ′ − x ) − f ( x ′ ) dx ′ ,where − R denotes the usual Cauchy principle value inte-gral), and under the parity transformation P : u ( x, t ) → u ( − x, t ) ≡ v ( x, t ), it changes to v t − vv x − Hv xx = 0.This mismatch of symmetry is paradoxical at first sight,but the paradox is resolved by interpreting u as a wavepropagating on one edge of a FQHE system and notingthat, in general, there is another edge far away carryinganother wave v . Thus, actually, the rational CMS modelcorresponds to two uncoupled BO equations for u and v . This system of equations is invariant under a paritytransformation interchanging u and v , P : [ u ( x, t ) , v ( x, t )] → [ v ( − x, t ) , u ( − x, t )] . (2)It is peculiar that these two BO equations are uncoupled,and it is for this reason that one can reduce the systemto a single equation, ignoring the other. While this un-coupling is reasonable if the two edges are infinitely farapart, it is natural to ask what would happen if the twoedges are parallel and close together; see Fig. 1. In thiscase, one would expect that the nonlinear waves propa-gating on the two edges interact. We now give a simpleheuristic argument to suggest that the hyperbolic CMSmodel can describe this situation. vu xL y FIG. 1. Schematic picture of a narrow FQHE system with twoedges carrying the two nonlinear waves u ( x, t ) and v ( x, t ). The hyperbolic CMS model can be defined by New-ton’s equations (1) with the interaction potential V ( r ) = X n ∈ Z r + 2i δn ) = (cid:16) π δ (cid:17) sinh − (cid:16) π δ r (cid:17) , (3) where δ > z j into two groups and shifting theones in the second group by the imaginary half-period, w k ≡ z k − N + i δ for k = 1 , . . . , N ≡ N − N , with1 < N < N , we can write these Newton’s equations as¨ z j = − N X j ′ = j V ′ ( z j − z j ′ ) − N X k =1 V ′ ( z j − w k ) , ¨ w k = − N X k ′ = k V ′ ( w k − w k ′ ) − N X j =1 V ′ ( w k − z j ) (4)for j = 1 , . . . , N and k = 1 , . . . , N , with˜ V ( r ) ≡ V ( r − i δ ) = − (cid:16) π δ (cid:17) cosh − (cid:16) π δ r (cid:17) . (5)This can be interpreted as a model of two kinds of par-ticles, z j and w k , in which particles of the same kindinteract via the singular repulsive two-body potential V , whereas particles of different kinds interact via theweakly attractive nonsingular potential ˜ V . We interpret δ as a parameter of the same order of magnitude as thedistance L y between the two edges of the FQHE system;see Fig. 1. In the rational limit δ → ∞ , we have ˜ V → u and v decouple; forfinite δ , the system is coupled. B. Nonchiral ILW equation.
In the hyperbolic case, the two-component generaliza-tion of the BO equation we present in this paper is givenby u t + 2 uu x + T u xx + ˜ T v xx = 0 ,v t − vv x − T v xx − ˜ T u xx = 0 (6)for u = u ( x, t ) and v = v ( x, t ), with( T f )( x ) ≡ δ − Z R coth (cid:16) π δ ( x ′ − x ) (cid:17) f ( x ′ ) dx ′ , ( ˜ T f )( x ) ≡ δ Z R tanh (cid:16) π δ ( x ′ − x ) (cid:17) f ( x ′ ) dx ′ . (7)The standard ILW equation is u t +2 uu x + T u xx = 0; it reduces to the BO equation in the limit δ → ∞ . Thus,if one drops the ˜ T -terms, (6) corresponds to a systemof uncoupled ILW equations generalizing the system ofuncoupled BO equations discussed above. However, dueto the presence of the ˜ T -terms, the nonlinear waves u and v interact. For this reason, and since equation (6) isinvariant under the parity transformation (2), we call itthe nonchiral ILW equation ; another motivation for thisname is its relation to a nonchiral conformal field theoryexplained in Section III A.For later reference, we also give the nonchiral versionof the periodic ILW equation: it is defined by (6) butwith the integral operators( T f )( x ) = 1 π − Z L/ − L/ ζ ( x ′ − x ) f ( x ′ ) dx ′ , ( ˜ T f )( x ) = 1 π Z L/ − L/ ζ ( x ′ − x + i δ ) f ( x ′ ) dx ′ , (8)where ζ ( z ) = πL lim M →∞ M X n = − M cot (cid:16) πL ( z − nδ ) (cid:17) (9)is equal to the Weierstrass elliptic ζ -function with periods( L, δ ), up to a term proportional to z . To see thatthe operators in (8) are natural periodic generalizationsof the ones in (7), we recall that π δ coth (cid:16) π δ z (cid:17) = lim M →∞ M X n = − M z − δn . (10) III. QUANTUM PHYSICS DESCRIPTION
It is known that the edge excitations in a FQHE sys-tem can be described by a conformal field theory (CFT)of chiral bosons, and that this CFT accommodates aquantum version of the BO equation which, at thesame time, provides a second quantization of the trigono-metric CMS system. This CFT is a nonlinear, ex-actly solvable system that can describe universal fea-tures of FQHE physics; in particular, as proposed byWiegmann, this description implies that the dynam-ics of FQHE edge states is essentially nonlinear, and itfeatures fractionally charged solitons with charges deter-mined by the filling level, ν .In this section, we explain how these results generalizeto the elliptic case, and how this led us to the nonchi-ral ILW equation (Section III A). We also substantiateour proposal that the (quantum version of the) nonchiralILW equation can describe the interaction of nonlinearwaves on two edges in a FQHE system, taking into ac-count interedge effects (Section III B). This section canbe skipped without loss of continuity. A. CFT and nonchiral ILW equation
The (quantum) elliptic CMS system is defined by theHamiltonian H N ( x ) = − N X j =1 ∂ ∂x j + X ≤ j
1) is to be interpreted as ( g − ~ ), i.e., g ( g − → g in the classical limit. Thus, for g = 2,the Hamiltonian in (11) defines the quantum analogue ofthe classical model defined by Newton’s equations in (1)for V ( x ) = ℘ ( x ). It is important to note that g is anessential parameter in the quantum case, different fromthe classical case where we can set g = 2 without loss ofgenerality. The CFT corresponding to the elliptic CMS systemcan be defined by two chiral boson operators ρ ( x ) (right-movers) and σ ( x ) (left-movers) labeled by a coordinate x ∈ [ − L/ , L/
2] on the circle with circumference
L > ρ ( x ) , ρ ( x ′ )] = − π i ν∂ x δ ( x − x ′ ) , [ σ ( x ) , σ ( x ′ )] = 2 π i ν∂ x δ ( x − x ′ ) , (13)and [ ρ ( x ) , σ ( x ′ )] = 0, with ν the filling factor of theFQHE system; the latter can be identified with theinverse of the coupling parameter in the correspondingCMS Hamiltonian: ν = 1 /g . For simplicity, we re-strict our discussion to FQHE states where g = 3 , , . . . ,even though the mathematical results discussed here holdtrue for arbitrary (rational) g > we use the subscript0 to distinguish these bare fields from dressed boson fields ρ ( x ) and σ ( x ) obtained from them by a Bogoliubov trans-formation, as described below.The linear dynamics of these fields is given by theHamiltonian (in this section and only here, we write R short for R L/ − L/ , to simplify notation) H = g π Z dx : (cid:16) ρ ( x ) + σ ( x ) + Z dx ′ h U ( x − x ′ )[ ρ ( x ) ρ ( x ′ )+ σ ( x ) σ ( x ′ )] − U ( x − x ′ ) ρ ( x ) σ ( x ′ ) i(cid:17) : (14)with colons indicating normal ordering and U j ( x ) = ∞ X n =1 q jn − q n cos(2 πnx/L ) ( j = 1 ,
2) (15)interaction potentials determined by the parameter q = e − πδ/L ( δ > . (16)The operator H is a special case of a Luttinger Hamil-tonian which, as is well-known, can be diagonalized by aBogoliubov transformation. This case is special in thatthe Bogoliubov transformed Hamiltonian has the sameform as for q = 0, except that the bare field operatorsare replaced by Bogoliubov transformed ones: H = g π Z dx : (cid:0) ρ + σ (cid:1) : . (17)This is a consequence of the special form of the inter-actions in (15), and it corresponds to the fact that theBogoliubov transformed fields ρ and σ provide two com-muting representations of the Virasoro algebra by theSugawara construction, as in the special case q = 0 wherethis is obvious; this is a manifestation of the fact thatwe are dealing with a nonchiral CFT (see e.g. Ref. [38and 39] for background on CFT). However, for nonzero q , the bare vacuum | i is not a highest weight state forthe dressed fields ρ and σ , and this has important conse-quences.The CFT described above accommodates the followingtwo kinds of vertex operators, φ ( x ) = : e − i g R x ρ ( x ′ )d x ′ : , ˜ φ ( x ) = : e i g R x σ ( x ′ )d x ′ : . (18)Moreover, using the boson operators above, one can con-struct a self-adjoint operator, H , providing a secondquantization of the elliptic CMS model in the followingsense: this operator satisfies the relations[ H , φ ( x ) · · · φ ( x N )] | i = H N ( x ) φ ( x ) · · · φ ( x N ) | i , (19)for arbitrary particle number N . We recently observed that it is possible to generalize H so that one has relations similar to the ones in (19)also for the vertex operators ˜ φ . This generalized oper-ator can be written as H = Z : (cid:20) g π (cid:0) ρ + σ (cid:1) + g ( g − π × (cid:0) ρT ρ x + σT σ x + ρ ˜ T σ x + σ ˜ T ρ x (cid:1)(cid:21) : dx (20)with the integral operators T , ˜ T in (8)–(9). Thus, theoperator H defines the following quantum version of theperiodic nonchiral ILW-equation,ˆ u t + 2 : ˆ u ˆ u x : + 12 ( g − T ˆ u xx + ˜ T ˆ v xx ] = 0 , ˆ v t − v ˆ v x : −
12 ( g − T ˆ v xx + ˜ T ˆ u xx ] = 0 . (21)To see this, we compute the Heisenberg equations of mo-tion A t = i[ H , A ] for A = ρ, σ and rescale, ρ → ˆ u ≡ gρ/ σ → ˆ v ≡ gσ/
2, to obtain (21). Moreover, by takingthe classical limit where the boson operators (ˆ u, ˆ v ) be-come functions ( u, v ) and ( g −
1) is replaced by g , andspecializing to g = 2, (21) reduces to (6). It is interesting to note that the operator in (20) satis-fies the following generalization of (19), allowing for bothkinds of vertex operators, φ and ˜ φ , at the same time:[ H , φ ( x ) · · · φ ( x N ) ˜ φ (˜ x ) · · · ˜ φ (˜ x N )] | i = H N ,N ( x , ˜ x ) φ ( x ) · · · ˜ φ (˜ x N ) | i (22)where H N ,N ( x , ˜ x ) = H N ( x ) + H N (˜ x )+ N X j =1 N X k =1 g ( g − ℘ ( x j − ˜ x k + i δ ) , (23)for arbitrary particle numbers N , N . This is a general-ization of the elliptic CMS Hamiltonian (11) describingtwo types of particles, where particles of the same typeinteract with the singular two-body potential ℘ ( x ), andparticles of different types interact with the nonsingu-lar attractive potential ℘ ( x + i δ ). It can be obtainedfrom a standard elliptic CMS Hamiltonian (11) by di-viding the particles into two groups and shifting the po-sitions in one group by i δ , similarly as in the classicalcase discussed in Section II A; see (4) ff . This argumentproves that the Hamiltonian H N ,N defines a quantumintegrable model. However, the physically relevant eigen-functions of H N ,N can not be obtained from the ones ofthe corresponding standard elliptic CMS Hamiltonian bythis shift trick. This suggests that the generalized modelcan describe new physics which would be interesting toexplore, but this is beyond the scope of the present paper.We finally mention that, to generate the full Hilbertspace of the CFT, one needs to consider two furtherkinds of vertex operators representing hole excitations,and there is a generalization of the result in (23) allow-ing for arbitrary numbers, N , M , N , M , of all fourtypes of vertex operators and with an interesting cor-responding Hamiltonian H N ,M ,N ,M , in generaliza-tion of a known result in the trigonometric case. Thus, H is actually the second quantization of these operators H N ,M ,N ,M generalizing the elliptic CMS Hamiltonian. B. Nonchiral CFT and FQHE
We motivate and explain our proposal that the nonchi-ral ILW equation can describe the interactions of nonlin-ear waves propagating on the two boundaries of a narrowFQHE system, in generalization of previous proposals forFQHE systems where the boundaries are well-separatedand interboundary interactions can be ignored. To pre-pare for this, we review known facts about the FQHE,bosonization and quantum hydrodynamics.
1. Projection to lowest Landau level
We recall the quantum mechanical description of acharged particle confined to the xy -plane in the presenceof a constant magnetic orthogonal to the plane (Lan-dau problem): Assuming periodic boundary conditionsin the x -direction: − L/ ≤ x ≤ L/ L = L x > y ∈ R , the exact eigenfunctions in the lowest Landaulevel (LLL) have the form ψ k ( x, y ) = e i kx e − ( y − k ) / , (24)using the Landau gauge and units where the magneticlength is set to 1, with k (short for k x ) an arbitrary inte-ger multiple of 2 π/L . In such a state, the particle has thebehavior of a plane-wave in the x -direction but is well-localized in the y -direction, and the quantum number k therefore has a two-fold physical interpretation: it can beinterpreted as momentum in x -direction and, at the sametime, it corresponds to the location of the wave packet in y -direction. As is well-known, the wave functions in theLLL are all degenerate: the energy is k -independent.We now consider the situation where, in addition to themagnetic field, we also have a potential, V conf ( y ), confin-ing the charged particle to a region − L y / ≤ y ≤ L y / L y >
0; this potential is zero at positions y far-ther away than some distance ℓ b > V conf ( y ) = 0 for | y ∓ L y / | > ℓ b , and it grows smoothly tovery large values in the boundary regions | y ∓ L y / | < ℓ b .In this situation, the degeneracy of the eigenfunctions inthe LLL is lifted, and the energy E ( k ) of the particleas a function of k is qualitatively similar to the function V conf ( y ) with y identified with k ; see Fig. 2. Thus, todescribe noninteracting such particles projected to theLLL, one can use the quantum many-body Hamiltonian H LLL = X k ( E ( k ) − µ ) ˆ ψ † ( k ) ˆ ψ ( k ) (25)with fermion field operators ˆ ψ ( † ) ( k ) obeying canonicalanticommutator relations, { ˆ ψ ( k ) , ˆ ψ † ( k ′ ) } = δ k,k ′ etc. ( µ is the chemical potential). By symmetry, we can assume E ( − k ) = E ( k ). y, k Energy E n ( k )loweredge upperedge E ( k ) E ( k ) E ( k ) µ − k F k F FIG. 2. Schematic picture of the lowest Landau level E ( k )in the presence of a potential confining the charged particlesto a region − L y / < y < L y /
2, as illustrated in Fig. 1. Thegrey lines indicate higher Landau levels that we ignore.
Thus, even though we consider a two dimensional sys-tem, it is modelled by a one-dimensional Hamiltonianthat can be treated by the bosonization method pio-neered by Haldane. This bosonized description is usefulsince it allows to find interactions that can be added to the Hamiltonian without spoiling integrability; as dis-cussed in the introduction, such interactions are particu-larly interesting in the context of FQHE physics.
2. Bosonization
We recall some pertinent facts about bosonization.
Consider the free fermion model defined by the Hamilto-nian (25). Its groundstate is the Dirac sea where all states − k F < k < k F are filled and all others are empty, withthe Fermi momentum k F > E ( k F ) = 0.It is convenient to decompose the (inverse) Fourier trans-form of the fermion field, ψ ( x ) = P k (2 π/L ) ˆ ψ ( k )e i kx , asfollows, ψ ( x ) = ψ + ( x )e i k F x + ψ − ( x )e − i k F x (26)with fermion field operators ψ ± ( x ) representing the low-energy excitations in the vicinity of the Fermi surfacepoints ± k F . As explained in Section III B 1, these Fermisurface points can be identified with the two boundaries, y = ± L y /
2, of a FQHE system, as illustrated in Fig. 1.The fermion fields on the RHS in (26) can be repre-sented by vertex operators, ψ ± ( x ) = : e ∓ i R x ρ ± ( x ′ )d x ′ : (27)where ρ ± ( x ) are operators satisfying the commutator re-lations of chiral bosons,[ ρ ± ( x ) , ρ ± ( x ′ )] = ∓ π i ∂ x δ ( x − x ′ ) (28)and [ ρ + ( x ) , ρ − ( x ′ )] = 0. These boson operators can beidentified with the corresponding fermion densities, ρ ± ( x ) = 2 π : ψ †± ( x ) ψ ± ( x ) : . (29)We note in passing that the boson fields ρ + ( x ) and ρ − ( x )are equal, up to a factor √ g and zero mode details, tothe bare boson fields ρ ( x ) and σ ( x ), respectively; seeSection III A.By Taylor-expanding the dispersion relation in thevicinity of the Fermi surface points: E ( ∓ k F + k ) = ± v F k + k m ∗ + . . . (30)with the Fermi velocity v F = E ′ ( k F ) and the effectivemass m ∗ = 1 /E ′′ ( k F ), one can expand H LLL = v F ( H (0)2 , + + H (0)2 , − ) + 1 m ∗ ( H (0)3 , + + H (0)3 , − ) + . . . (31)with H (0)2 , ± = 14 π Z : ρ ± ( x ) : dx, H (0)3 , ± = 112 π Z : ρ ± ( x ) : dx (32)etc. This provides a basis for the quantum hydrody-namic description of such systems proposed by Abanovand Wiegmann.
3. Chiral Luttinger liquids and FQHE
We recall Wen’s chiral Luttinger liquid description ofFQHE systems. The leading term in (31), H (0)2 = v F ( H (0)2 , + + H (0)2 , − ) , (33)provides a good starting point to describe FQHE sys-tems, but the low-energy excitations are not fermionsbut rather collective excitations that can be describedby vertex operators φ ± ( x ) = : e ∓ i √ g R x ρ ± ( x ′ )d x ′ : (34)with g = 3 , , . . . at filling level ν = 1 /g ; the fermion case g = 1 corresponds to the integer Hall effect and, for g > one boundary or, equivalently, to one chiral sector, + or − .This is Wen’s chiral Luttinger liquid model.
4. Boundary waves in FQHE systems
The dynamics of the boson fields provided by theHamiltonian H (0)2 via the Heisenberg equations of mo-tion is ∂ t ρ ± ± v F ∂ x ρ ± = 0 . (35)These linear equations describe waves propagating at thetwo boundaries of a FQHE system: at each boundary,the wave packets move in one direction, right (+) or left( − ), with constant speed v F and without changing shape.The Hamiltonian H (0)2 is highly degenerate, and it isnatural to ask if one can lift this degeneracy by addinginteractions that fulfill the following requirements: (i)they do not spoil integrability, (ii) they provide nonlin-ear corrections to the linear wave equations (35), (iii)they are compatible with the vertex operators (34) de-scribing composite fermions. An interesting Hamilto-nian obtained by adding such terms to H (0)3 , ± (32) is H , ± = Z : (cid:16) √ g π ρ ± ( x ) + g − π ρ ± H ( ρ ± ) x (cid:17) : dx (36)with the Hilbert transform H (obtained from T in (8)by taking the limit δ → ∞ ): the dynamics for the bosonfields provided by this Hamiltonian is a quantum ver-sion of the BO equation which is integrable, and thisHamiltonian is compatible with the composite fermionoperators in that it also provides a second quantizationof trigonometric CMS model; using the latter and theknown eigenfunctions of the trigonometric CMS system,one can construct the exact eigenstates of H , ± .
5. Proposal
We now are ready to motivate and explain our pro-posal that the nonchiral ILW equation can describe wavespropagating on parallel boundaries of FQHE systems.We recall that, in generic applications of bosoniza-tion, the most important interactions to be added tothe Hamiltonian H (0)2 are quadratic in the boson op-erator, and thus, generically, one obtains a LuttingerHamiltonian as in (14), for some potentials, U ( x ) and U ( x ); these potentials describe interactions betweenthe same ( U ) and opposite ( U ) chiral degrees of free-dom. Moreover, one often assumes that these interac-tions are local since this guarantees that the resultingmodel is conformally invariant. In the context of theFQHE, such Luttinger interactions are usually ignoredby the following arguments: (i) the two chiral degrees offreedom describe excitations at two separated boundariesof the system, and U ( x ) therefore describes interedge in-teractions which are negligible if the boundaries are suffi-ciently far apart; (ii) a local interaction U ( x ) within thesame boundary only renormalizes the Fermi velocity andthus can be taken into account by redefining v F . How-ever, since the one-particle eigenfunctions of the LandauHamiltonian are spatially extended, and Coulomb inter-actions in a FQHE system are long-range, there is noreason to exclude nonlocal interactions which preserveconformal symmetry. Moreover, it is known that trans-port coefficients in Luttinger liquid are universal even ifthe interactions mix the chiral degrees of freedom and arenonlocal, i.e., the accurate quantization of the Hall con-ductance observed in real FQHE systems is compatiblewith generic Luttinger model interactions; see Ref. [44]for a recent construction of the pertinent general Lut-tinger model for general vertex operators as in (34).As discussed in Section III A, the Luttinger Hamilto-nian (14) with the fine-tuned interactions in (15) is con-formally invariant, for arbitrary fixed δ >
0, and thereare natural corresponding generalizations of the compos-ite fermion operators and the operator in (36) satisfy-ing the requirements stated in Section III B 4: they aregiven in (18) and (20), respectively. Our proposal tomodel a FQHE system at filling 1 /g , g = 3 , , . . . , andwith the geometry illustrated in Fig. 1 is therefore as fol-lows: The boson field operators ρ and σ , satisfying thecommutator relations in (13) , describe low-energy excita-tions located at the upper and lower edge, respectively, ofthe FQHE system boundary; the low-energy descriptionof the system is by the Hamiltonian H in (14) – (15) ,with the parameters v F and δ determined by system de-tails like the edge distance, L y , and the confining poten-tial, V conf ( y ) ; the linear- and nonlinear dynamics of theboundary waves is described by the operators in (14) and (20) , respectively; the vertex operators in (18) describequasiparticle excitations in the system.
6. Inter-edge effects in FQHE systems
We argue that the model proposed in Section III B 5can describe interedge effects in narrow FQHE system.Our model predicts that the quasiparticles of the sys-tem are the Bogoliubov transformed boson fields, ρ and σ , diagonalizing H in (14); see (17). Thus, the lineardynamics is given by the same equations as for δ = ∞ ,i.e., ρ t − v F ρ x = 0 and σ t + v F σ x = 0. However, since ρ (say ) is a superpositions of the fields ρ and σ localizedat two distinct boundaries, a right-moving wave excitedat the upper edge will always develop into a pair of well-defined corresponding excitations at both edges movingin parallel. Thus, our proposal can be tested already inexperiments on real FQHE systems that can only resolve linear boundary waves: our model predicts correspond-ing excitations, u ( x − v F t ) and v ( x − v F t ) proportionalto the expectation values of ρ ( x, t ) and σ ( x, t ), respec-tively, where u ( x ) and v ( x ) are determined by a singlefunction, u ( x ), and the inverse of the Bogoliubov trans-formation described in Section III A ( u ( x ) is proportionalto the expectation value of ρ ( x, t = 0)). Furthermore, ourmodel predicts nonlinear waves described by the quan-tum nonchiral ILW equation (21), in generalization ofthe Wiegmann proposal quoted in the beginning of thissection. It would be interesting to elaborate these pre-dictions in detail, and to propose specific experimentson real FQHE systems to test them. Clearly, this is aresearch project in its own. Our results in Sections IVand (V) are a first step, giving an indication of the newphysics that the nonchiral ILW equation can describe.To elaborate predictions of our model, it would be in-teresting to construct the exact eigenstates of the Hamil-tonian H in (20), in generalization of known results for δ = ∞ . This is challenging. One reason is that, whilethe exact eigenstates of the trigonometric CMS modelhave been known for a long time, the ones of the rele-vant elliptic CMS-type systems are the subject of ongoingresearch. IV. RESULTS: HYPERBOLIC CASEA. Multisoliton solutions
The following fundamental result shows that (6) ad-mits multisoliton solutions whose dynamics is describedby the hyperbolic CMS model, thus generalizing a fa-mous result for the rational case: For arbitrary integers N ≥ and complex parameters a j with imaginary partsin the range δ/ < Im a j < δ/ for j = 1 , . . . , N , the fol-lowing is an exact solution of the nonchiral ILW equation (6) : (cid:18) u ( x, t ) v ( x, t ) (cid:19) = i N X j =1 (cid:18) α ( x − z j ( t ) − i δ/ − α ( x − z j ( t ) + i δ/ (cid:19) +c . c . (37) where α ( x ) = ( π/ δ ) coth( πx/ δ ) and the poles z j ( t ) aredetermined by Newton’s equations (1) with V ( r ) given by (3) and with initial conditions z j (0) = a j and ˙ z j (0) = 2i N X j ′ = j α ( a j − a j ′ ) − N X k =1 α ( a j − a k + i δ ) (38)(the bar denotes complex conjugation, c.c.). Thus, toobtain an exact solution of (6), one chooses complex pa-rameters a j satisfying δ/ < Im a j < δ/
2; next, thetime-evolution of z j ( t ) is obtained by solving the hyper-bolic CMS model with initial conditions determined bythe a j ; finally, the solution of (6) is obtained from (37).Using the exact analytic solution of the hyperbolic CMSmodel obtained by the projection method, the numeri-cal effort to compute such an multisoliton solution at anarbitrary time, t , is reduced to diagonalizing an explicitlyknown N × N matrix. As elaborated in Appendix B, wetested this result by comparing with a numeric solutionof (6). u , v x FIG. 3. Time evolution of a two-soliton solution of the nonchiral ILW equation (6) with a u -channel dominated soliton (bigblue and small red humps) colliding with a v -channel dominated soliton (big red and small blue humps), as explained in themain text. The plots show u ( x, t ) (blue line) and v ( x, t ) (red line) at successive times t = ( n − t , n = 1 , . . . ,
5; the parametersare δ = π , a = − . δ , a = 3 + 0 . δ , and t = 2 . B. Examples.
The one-soliton solution of (6) is given by (cid:18) u ( x, t ) v ( x, t ) (cid:19) = i (cid:18) α ( x − z ( t ) − i δ/ − α ( x − z ( t ) + i δ/ (cid:19) + c . c ., (39) where the poles evolve linearly in time, with initial con-ditions determined by a complex parameter a such that y t/t x x FIG. 4. (a) Time evolution of the poles z j ( t ), j = 1 ,
2, inthe complex plane corresponding to the two-soliton solutionin Fig. 3. The times t = ( n − t , n = 1 , . . . ,
5, defined inthe caption of Fig. 3 are indicated by circles; the arrows markcircles corresponding to n = 1. The dotted lines indicate theevolution of poles without interactions. (b) Time evolution ofthe center-of-mass locations of the solitons given by Re z j ( t ). δ/ < Im a ≤ δ/ z ( t ) = a + ˙ z (0) t, ˙ z (0) = 2i α ( a − ¯ a + i δ ) . (40)It is important to note that ˙ z (0) is real, and therefore,Im z ( t ) = Im a independent of t . Thus, the functions u ( x, t ) and v ( x, t ) both describe humps whose shapesdo not change with time. These humps are centeredat the same point and move with constant velocity,Re z ( t ) = Re a + ˙ z (0) t , and their heights, max u > v >
0, are determined by Im a . For Im a close to 3 δ/
2, max u ≫ max v , and the solitons moveto the right, ˙ z (0) >
0. As Im a decreases, max u and˙ z (0) decrease while max v increases until, at Im a = δ ,max u = max v and ˙ z (0) = 0. Thus, if Im a lies in therange δ < Im a < δ/
2, then the one-soliton is mainlyin the u -channel and moves to the right; it is there-fore similar to the one-soliton solution of the standardILW equation u t + 2 uu x + T u xx = 0. Similarly, when δ/ < Im a < δ , the one-soliton is mainly in the v -channeland moves to the left, similar to a one-soliton solution ofthe P -transformed ILW equation v t − vv x − T v xx = 0.For parameters a j such that Re( a j − a k ) ≫ δ forall j = k , the multisoliton solution of (6) is well-approximated by a sum of N one-solitons of the form(39) where ˙ z j ( t ) ≈ α ( a j − ¯ a j + i δ ) is time-independentfor times such that Re( z j ( t ) − z k ( t )) ≫ δ ; see Fig. 3 fora two-soliton solution, with the corresponding motion ofpoles in Fig. 4(a). However, when two solitons meet, theyinteract in a nontrivial way, and after the interaction theyre-emerge with the same shape but with phase-shifts; seeFig. 4(b). Such nontrivial interactions between solitonscan also be modeled by the system of decoupled ILWequations obtained from (6) by dropping the ˜ T -terms.A qualitatively new effect stemming from the ˜ T -terms isthat u -channel dominated solitons ( u -solitons) interactnontrivially with v -solitons, as clearly seen in our exam-ple in Figs. 3 and 4. It is interesting to note that thepoles corresponding to the u - and v -solitons interchangetheir imaginary parts and directions during the collisionand thus, in this sense, exchange their identities: whilethe first pole corresponds to the u -soliton and the sec-ond to the v -soliton before the collision, it is the other way round after the collision; see Figs. 4(a) and (b). Wenote that such an identity change of poles during solitoncollisions is known for the BO equation, but only forsolitons moving in one direction. C. Derivation of multisoliton solutions.
We explain the key difference between the derivationof solitons for (6) and the corresponding derivation inthe rational case; further details can be found in Ap-pendix A 1.The Hilbert transform, H , satisfies H = − I , and thisproperty is crucial for the existence of eigenfunctions of H needed in the derivation of the CMS-related solitonsolutions of the BO equation u t + 2 uu x + Hu xx = 0. However, while the trigonometric generalization of H alsohas this property, the hyperbolic generalization of H isthe operator T in (7), and T = − I . This is the reasonwhy the soliton solution of the BO equation straightfor-wardly generalizes to the trigonometric case, but thenaive generalization to the hyperbolic case fails. How-ever, the nonchiral ILW equation (6) can be written invector form as u t + ( u . u ) x + T u xx = 0 , u ≡ (cid:18) uv (cid:19) , u . u ≡ (cid:18) u − v (cid:19) , T ≡ (cid:18) T ˜ T − ˜ T − T (cid:19) (41)where the matrix operator, T , satisfies T = − I . More-over, ( α ( x + z ± i δ/ , − α ( x + z ∓ i δ/ t are eigenfunctionsof T with eigenvalues ± i. The latter are the eigenfunc-tions needed to be able to use the method developed forthe rational case: using well-known identities for thefunction α ( x ), as well as a B¨acklund transformationfor the hyperbolic CMS model, it is straightforward toadapt a known derivation of multisoliton solutions of theBO equation to the hyperbolic case. D. Integrability.
We found a Lax pair, a Hirota bilinear form, aB¨acklund transformation, and an infinite number of con-servation laws for (6). Thus, the nonchiral ILW equationis a soliton equation that is integrable in the same strongsense as the standard ILW equation. Below we presentsome of these results that can be checked by straightfor-ward computations.The Lax pair we found is as follows:
Let ψ ( z ; t, k ) be ananalytic function on the union of the strips < Im z < δ and δ < Im z < δ and extended to C by δ -periodicity, ψ ± ( x ; t, k ) and ψ ± δ ( x ; t, k ) the boundary values of thisfunction on R and R + i δ , respectively, and µ , µ , ν ,and ν arbitrary functions of the spectral parameter k .Then the compatibility of the following linear equations yields (6) : (i ∂ x − u − µ ) ψ − = ν ψ +0 , (i ∂ x + v − µ ) ψ + δ,x = ν ψ − δ , (cid:16) i ∂ t − µ i ∂ x − ∂ x + T u x + ˜ T v x ± i u x + µ (cid:17) ψ ± = 0 , (cid:16) i ∂ t − µ i ∂ x − ∂ x + T v x + ˜ T u x ± i v x + µ (cid:17) ψ ± δ = 0 . Inspired by known results for the BO equation, weobtained the following Hirota bilinear form of (6),(i D t − D x ) F − · G + = (i D t − D x ) F + · G − = 0 (42)with u = i ∂ x log( F − /G + ) and v = − i ∂ x log( F + /G − ),where F ± ( x, t ) ≡ F ( x ± i δ/ , t ) and similarly for G , usingstandard Hirota derivatives. The first three of the conservation laws we found are I = Z R ( u + v )d x, I = 12 Z R ( u − v )d x,I = Z R (cid:20) u uT u x u ˜ T v x u ↔ v ) (cid:21) d x (43)with ( u ↔ v ) short for the same three terms but with u and v interchanged.B¨acklund transformations, other conservation laws,and detailed derivations are given elsewhere. V. RESULTS: ELLIPTIC CASE
To generalize (6) to the periodic setting, we usethe Weierstrass functions ℘ ( z ) and ζ ( z ) with periods(2 ω , ω ) ≡ ( L, δ ), L >
0, and the related functions ζ j ( z ) ≡ ζ ( z ) − η j z/ω j , η j ≡ ζ ( ω j ), j = 1 ,
2. The function ζ ( z ) is L -periodic, ζ ( z + L ) = ζ ( z ), whereas the func-tion ζ ( z ) is 2i δ -periodic, ζ ( z + 2i δ ) = ζ ( z ); recall that ζ ( z ) is neither L - nor 2i δ -periodic. We note that ℘ ( x )in (12) equals − ζ ′ ( x ) = ℘ ( x ) + η /ω .The periodic nonchiral ILW equation is given by (6)with the integral operators T , ˜ T in (8)–(9). With that, T in (41) still satisfies T = − I , and the derivation of themultisoliton equation outlined above generalizes straight-forwardly to the elliptic case provided α ( z ) in (A9) ischosen as the 2i δ -periodic variant of ζ ( z ): The functions u ( x, t ) and v ( x, t ) given in (37) , with α ( x ) = ζ ( z ) , sat-isfy the periodic nonchiral ILW equation provided that z j ( t ) satisfy Newton’s equations (1) with the elliptic CMSmodel potential V ( r ) = ℘ ( r ) , and with initial conditions z j (0) = a j and ˙ z j (0) in (38) , for arbitrary complex a j satisfying δ/ < Im a < δ/ and − L/ ≤ Re a j < L/ , j = 1 , . . . , N . It is important to note that the multisoli-ton solution is L -periodic even though ζ ( z ) is not. Theinterested reader can find further details in Appendix A 2. VI. OTHER APPLICATIONS
We present arguments suggesting that the nonchiralILW equation introduced in this paper will find other ap-plications in physics beyond the application to the FQHE described earlier (Section VI A). As a specific example,we discuss a possible application in the context of nonlin-ear water waves, and thereby provide a complementaryphysical interpretation of our mathematical results (Sec-tion VI B).
A. The wide applicability of soliton equations
Nonlinear evolution equations are typically more dif-ficult to solve than linear ones, and theoretical physicstools are often not equally powerful when nonlinear ef-fects are important. Soliton equations are an impor-tant exception: these nonlinear equations are integrable,and it is therefore possible to develop analytic andnumeric methods to solve them reliably. Thus, phe-nomena described by soliton equations can be very wellunderstood despite of the crucial importance of nonlin-ear effects. The class of such phenomena is remarkablylarge, with many examples from different areas in physicssuch as hydrodynamics, nonlinear optics, plasma physics,dislocation theory of crystals, etc. A well-known expla-nation of this wide applicability of soliton equations isby Calogero: certain “universal” nonlinear PDEs canbe obtained, by a limiting procedure involving rescalingsand an asymptotic expansion, from very large classes ofnonlinear evolution equations [. . . ]. Because this limitingprocedure is the correct one to evince nonlinear effects,the universal model equations obtained in this manner[. . . ] are widely applicable. Because this limiting pro-cedure generally preserves integrability, these universalmodel equations are likely to be integrable [. . . ]. This suggests that the nonchiral ILW (6) will find otherapplications in physics.
B. Nonlinear water waves
Consider the following class of soliton equations de-scribing, e.g., nonlinear water waves in different situa-tions: u t + 2 uu x + Du xx = 0 (44)where D is one of the linear operators specified be-low and u = u ( x, t ), where x is a coordinate on one-dimensional space and t time. This class includes thefamous Korteweg-de Vries (KdV) equation, the BOequation, the ILW equation interpolating betweenthe KdV and the BO equations, and periodic vari-ants of these three equations depending on a furtherparameter, L >
0, corresponding to the spatial period: u ( x + L, t ) = u ( x, t ).While the nonlinear term, 2 uu x , is the same in allcases, the dispersive term, Du xx , is different: it amountsto multiplication of u by functions iΩ( k ) in Fourier space: Du xx = iΩ( − i ∂ x ) u , with the following dispersion rela-0tions in the different cases, Ω( k ) = k δ/ k sgn( k ) (BO) k coth( kδ ) (ILW) (45)where the wave number, k , is restricted to integer multi-ples of 2 π/L in the periodic cases (it is real otherwise),and δ > D is represented by a differential operatorin the KdV case, Df = δ∂ x f /
3, whereas in the BO- andILW cases it is given by an integral operator denoted as H (Hilbert transform) and T , respectively; see (A3).It is important to note that, in general, one should adda term cu x to the LHS in (44), with c some velocity pa-rameter, to make manifest that (44) is a generalizationof the chiral wave equation u t + cu x = 0; however, sincethis term is trivial in that it can be removed by a trans-formation u → u − c/
2, we ignore it in our discussion.The soliton equations in (44)–(45) provide effective de-scriptions of nonlinear water waves taking into accountthe most important nonlinear and dissipative terms. It is important to note that, when deriving these equa-tions from fundamental hydrodynamic laws, parity in-variance is broken and, for this reason, the equationsin (44)–(45) are chiral : they can only describe solitonsmoving to the right. Obviously, one can obtain a corre-sponding equation describing solitons moving to the leftby a parity transformation: v ( x, t ) ≡ u ( − x, t ) satisfies v t − vv x − Dv xx = 0. Thus, the chiral equation in (44)–(45) actually corresponds to a system of two equationsfor u and v describing solitons moving in both directions.Clearly, this description is simplistic with regard to thefollowing: solitary waves in nature moving in oppositedirections interact when they meet, but such interactionsare ignored by this uncoupled system for u and v . Thissuggests to try to find integrable generalizations of theseequations of the form u t + 2 uu x + Du xx + X ( v, u ) = 0 ,v t − vv x − Dv xx − X ( u, v ) = 0 (46)with coupling terms, X ( v, u ) and X ( u, v ), such that thesystem (46) is invariant under the parity transformationin (2). We believe that neither the KdV equation northe BO equation allow for such a coupling; however, theILW equation does: it is given by the dispersive term I ( v, u ) = ˜ Dv xx = i ˜Ω( − i ∂ x ) v (independent of u ) with˜Ω( k ) = k sinh( kδ ) ; (47)indeed, using (A3), one sees that (46) in this case is equiv-alent to the nonchiral ILW equation (6) ff .One can check that (6) does not have a well-definedlimit δ →
0, and that ˜
T u xx → δ → ∞ : theKdV-limit of the nonchiral ILW equation does not exist,and its BO-limit is trivial. Thus, to describe nonchiralphysics, one has to work in the regime 0 < δ < ∞ . This suggests that it would be interesting to revisit thederivation of the KdV-equation from more fundamentalparity invariant equations, and to see if this can be gen-eralized so as to obtain the nonchiral ILW equation. VII. FINAL REMARKS
We presented the novel soliton equation (6). We callit the nonchiral ILW equation because it is parity invari-ant and can describe interacting solitons moving in bothdirections. We obtained exact multisoliton solutions de-termined by poles satisfying the equations of motion ofthe hyperbolic CMS model, and we gave a Lax pair, aHirota form, and conservation laws. We also presented aperiodic nonchiral ILW equation and its soliton solutionsdetermined by the elliptic CMS model.We proposed that the nonchiral ILW equation canmodel coupled nonlinear waves in FQHE systems, andwe gave background information to make this proposalprecise. However, as we argued, our results are of widerinterest: Many soliton equations containing only first-order derivatives in time are chiral, i.e., they can onlydescribe solitons moving in one direction, left or right,and thus are not parity invariant. Examples include theKdV equation, the BO equation and, more generally, theILW equation. However, the fundamental equations inhydrodynamics from which these soliton equations arederived are parity invariant. This mismatch of symme-tries is not fully satisfactory. Using the nonchiral ILWequation instead of the standard ILW equation recon-ciles symmetries, and we therefore believe that, in var-ious applications in physics, the former can be a betterapproximation to fundamental equations than the latter.We hope that our results open up a route to generalizerecent results on a generalized hydrodynamic descriptionof the Toda chain to the elliptic CMS model. Thiswould be interesting since, in the elliptic CMS model, onecan change the qualitative character of the interactionfrom long-range in the trigonometric case, to short-rangein the hyperbolic case, to nearest-neighbor in the Todalimit.
ACKNOWLEDGMENTS
We thank L. Bystricky, M. Noumi, and in particularJ. Shiraishi for very helpful and inspiring discussions.B.K.B. acknowledges support from the G¨oran GustafssonFoundation and the European Research Council, GrantAgreement No. 682537. E.L. acknowledges support bythe Swedish Research Council, Grant No. 2016-05167,and by the Stiftelse Olle Engkvist Byggm¨astare, Contract184-0573. J.L. acknowledges support from the G¨oranGustafsson Foundation, the Ruth and Nils-Erik Stenb¨ackFoundation, the Swedish Research Council, Grant No.2015-05430, and the European Research Council, GrantAgreement No. 682537.1
Appendix A: Derivation of soliton solutions
We give details on the derivation of the N -soliton solu-tions presented in the main text, both in the hyperbolicand elliptic cases.
1. Hyperbolic case
We construct solutions of (41) with T, ˜ T definedin (7) by generalizing a known method for the BOequation. a. Integral operators in Fourier space We compute the Fourier space representation of thematrix operator T in (41).We start by transforming the operators T, ˜ T in (7) toFourier space, using the following exact integral, Z R π δ coth (cid:16) π δ ( x ∓ i a ) (cid:17) e − i kx dx = − π i e ± ( ak − kδ ) sinh( kδ ) (A1)for real parameters a, k such that 0 < a < δ and k = 0(a derivation of this result can be found at the end of thissection). This implies − Z R δ coth (cid:16) π δ x (cid:17) e − i kx dx = − i coth( kδ ) , Z R δ tanh (cid:16) π δ x (cid:17) e − i kx dx = − i 1sinh( kδ ) (A2)for real k = 0. Indeed, the first of these identities isequivalent to the average of the two integrals in (A1) inthe limit a ↓
0, and the second is obtained from (A1)in the special case a = δ . Observe that the integralsin (7) are convolutions. Using the following conventionsfor Fourier transformation, ˆ u ( k ) = R R u ( x )e − i kx dx , theoperators defined in (7) can therefore be expressed inFourier space as follows, [ ( T u )( k ) = i coth( kδ )ˆ u ( k ) , [ ( ˜ T u )( k ) = i 1sinh( kδ ) ˆ u ( k ) . (A3)Thus, for the matrix operator T defined in (41), d T u ( k ) =ˆ T ( k )ˆ u ( k ) withˆ T ( k ) = i (cid:18) coth( kδ ) 1 / sinh( kδ ) − / sinh( kδ ) − coth( kδ ) (cid:19) (A4)and ˆ u ( k ) = (ˆ u ( k ) , ˆ v ( k )) t for u ( x ) = ( u ( x ) , v ( x )) t . Us-ing this, it is easy to check that ˆ T ( k ) = − I , which isequivalent to T = − I . Derivation of (A1) . Suppose 0 < a < δ and define thefunction h ( x ) by h ( x ) = π δ coth (cid:16) π δ ( x − i a ) (cid:17) . Even though h ( x ) does not decay as x → ±∞ , the Fouriertransform ˆ h of h is well-defined as a tempered distribu-tion. Indeed, the derivative h ′ ( x ) = − (cid:18) π δ sinh( π ( x − i a )2 δ ) (cid:19) has exponential decay as x → ±∞ and has a double poleat x = i a + 2i δn for each integer n . Its Fourier transform d ( h ′ ) can be computed by a residue computation. TheFourier transform ˆ h can then be obtained for k = 0 byˆ h ( k ) = d ( h ′ )( k ) / (i k ). A similar computation applies if − δ < a <
0, and we arrive at (A1). b. Eigenfunctions
Since T = − I , the eigenvalues of T are ± i. We nowconstruct the corresponding eigenfunctions.By straightforward computations we obtain the follow-ing eigenvectors of the matrix ˆ T ( k ) in (A4),ˆ g ( k ) (cid:18) e ± kδ/ − e ∓ kδ/ (cid:19) (A5)with corresponding eigenvalues ± i, for an arbitrary func-tion ˆ g ( k ) of k . To get eigenfunctions of T with appropri-ate analyticity properties, we restrict ourselves to func-tions ˆ g ( k ) such that ˆ g ( k )e kα has a well-defined inverseFourier transform g ( x − i α ) in a strip − A < α < A with
A > δ/
2. For such functions, Z R dk π ˆ g ( k )e ± kδ/ e i kx = g ( x ∓ i δ/ , (A6)and the eigenfunctions of the operator T are therefore asfollows: For arbitrary complex valued functions g ( z ) of z ∈ C analytic in a strip − A <
Im( z ) < A with A > δ/ ,the vector valued functions v ± ( x ) ≡ (cid:18) g ( x ∓ i δ/ − g ( x ± i δ/ (cid:19) (A7) satisfy T v ± ( x ) = ± i v ± ( x ) . (A8) c. Pole ansatz Inspired by the CMS-related soliton solutions knownfor the BO equation, we make the following ansatz2to solve (41), u ( x, t ) = i N X j =1 (cid:18) α ( x − z j ( t ) − i δ/ − α ( x − z j ( t ) + i δ/ (cid:19) − i M X j =1 (cid:18) α ( x − w j ( t ) + i δ/ − α ( x − w j ( t ) − i δ/ (cid:19) (A9)where α ( x ) = ( π/ δ ) coth( πx/ δ ), N, M are arbitraryintegers ≥
0, and with poles z j ( t ) and w j ( t ) to be de-termined. We note that, to obtain real-valued solutions,one must restrict this ansatz to (37), i.e., M = N and w j ( t ) = ¯ z j ( t ) for all j , but we find it convenient to derivea more general result. In the following, we sometimeswrite z j as shorthand for z j ( t ), etc.The function α ( z ) is meromorphic with poles at z =2i δn , n integer. Thus, if we restrict the imaginary partsof z j and w j as follows,Im( z j ± i δ/ = 2 δn, Im( w j ± i δ/ = 2 δn (A10)for all integers n , then the result in (A7)–(A8) implies T u xx = − N X j =1 (cid:18) α ′′ ( x − z j − i δ/ − α ′′ ( x − z j + i δ/ (cid:19) − M X j =1 (cid:18) α ′′ ( x − w j + i δ/ − α ′′ ( x − w j − i δ/ (cid:19) (A11)with α ′ ( z ) ≡ ∂ z α ( z ) etc. We now use α ( − z ) = − α ( z )and the well-known identities α ′ ( z ) = − V ( z ) , ∂ z (cid:2) α ( z ) (cid:3) = V ′ ( z ) ,α ( z + 2i δ ) = α ( z ) , ∂ z (cid:2) α ( z − a ) α ( z − b ) (cid:3) = ∂ z (cid:2) α ( z − a ) − α ( z − b ) (cid:3) α ( a − b ) , (A12)with V in (3), and for arbitrary z, a, b ∈ C . Using thiswe compute u t + ( u . u ) x + T u xx = N X j =1 (cid:18) V ( x − z j − i δ/ − V ( x − z j + i δ/ (cid:19) × i ˙ z j + 2 N X k = j α ( z j − z k ) − M X k =1 α ( z j − w k + i δ ) + M X j =1 (cid:18) V ( x − w j + i δ/ − V ( x − w j − i δ/ (cid:19) × − i ˙ w j + 2 M X k = j α ( w j − w k ) − N X k =1 α ( w j − z k + i δ ) (the computations leading to this result are nearly thesame as in the BO case and thus omitted). This impliesthe following result: The function in (A9) satisfies the nonchiral ILW equation in (41) provided the followingsystem of equations is satisfied, ˙ z j = 2i N X k = j α ( z j − z k ) − M X k =1 α ( z j − w k + i δ ) , ˙ w j = − M X k = j α ( w j − w k ) + 2i N X k =1 α ( w j − z k + i δ ) , (A13) and the conditions in (A10) hold true. The system in (A13) is known as a
B¨acklund transfor-mation for the hyperbolic CMS system. It implies twodecoupled systems of Newton’s equations,¨ z j = − N X k = j V ′ ( z j − z k ) ( j = 1 , . . . , N ) , (A14a)¨ w j = − M X k = j V ′ ( w j − w k ) ( j = 1 , . . . , M ) (A14b)with V as in (3); see Ref. [61] for a recent alternativederivation of this result. We thus obtain the followinggeneralization of the result stated in the main text: Forarbitrary nonnegative integers
N, M and complex param-eters a j , j = 1 , . . . , N , and b j , j = 1 , . . . , M , satisfying Im( a j ± i δ/ = 2 δn, Im( b j ± i δ/ = 2 δn (A15) for all integers n , the function u ( x, t ) in (A9) is a so-lution of the nonchiral ILW equation (41) provided thepoles z j ( t ) and w j ( t ) satisfy Newton’s equations for thehyperbolic CMS model in (A14) with initial conditions z j (0) = a j , w j (0) = b j , ˙ z j (0) = 2i N X k = j α ( a j − a k ) − M X k =1 α ( a j − b k + i δ ) , ˙ w j (0) = − M X k = j α ( b j − b k ) + 2i N X k =1 α ( b j − a k + i δ ) . Restricting to M = N and b j = ¯ a j for all j , we obtain theresult stated in the main text (note that, in this specialcase, the initial conditions imply w j ( t ) = ¯ z j ( t ) for all t ).A technical remark is in order. Strictly speaking, weproved the result above only for times, t , where the con-ditions in (A10) hold true. We did not point out thisrestriction before since we believe that, if the conditionsin (A10) and (A13) hold true at time t = 0, then thesolutions z j ( t ) and w j ( t ) of (A14) satisfy the conditionsin (A10) for all t >
0. We checked this in several specialcases by integrating (A14) numerically. We expect thatthis can be proved in general using the known explicitsolution of the hyperbolic CMS model obtained with theprojection method; this is left for future work.3
2. Elliptic case
We give details on how the derivation in Appendix A 1generalizes to the L -periodic case. a. Periodic nonchiral ILW equation To see that (6) with T , ˜ T in (8)–(9) is the correct L -periodic generalization of the nonchiral ILW equation,one can check that (A3) still holds true but with Fouriermodes, k , restricted to integer multiples of (2 π/L ), andfor L -periodic functions f ( x ) that have zero mean, ˆ f (0) ≡ R L/ − L/ f ( x ) dx = 0. Thus, T = − I , and the result in(A7)–(A8) holds true as it stands provided the function f ( z ) is L -periodic, has zero mean, and is analytic in astrip − A <
Im( z ) < A for A > δ/
2. In particular, T ∂ x (cid:18) ζ ( x − z ∓ i δ/ − ζ ( x − z ± i δ/ (cid:19) = ∓ i (cid:18) ℘ ′ ( x − z ∓ i δ/ − ℘ ′ ( x − z ± i δ/ (cid:19) (A16)using ζ ′′ ( z ) = − ℘ ′ ( z ). We can use this to construct soli-ton solutions related to the elliptic CMS model definedby Newton’s equations (1) with the potential V ( x ) = ℘ ( x ) . (A17) b. Pole ansatz The discussion above suggests to use the pole ansatz in(A9) with α ( x ) equal to ζ ( x ). However, this choice doesnot work since the third identity in (A12) is not satisfied.The choice that works is α ( x ) = ζ ( x ) (A18)since ζ ( z ) is 2i δ -periodic. However, ζ ( z ) is not L -periodic: ζ ( z + L ) = ζ ( z ) + c for some nonzero constant c . Thus, u ( x + L, t ) = u ( x, t ) + i( N − M )( c, − c ) t , and,to get a L -periodic function u ( x, t ), we must restrict to M = N .We use (A16) to obtain T u xx = N X j =1 (cid:18) V ′ ( x − z j − i δ/ − V ′ ( x − z j + i δ/ (cid:19) + N X j =1 (cid:18) V ′ ( x − w j + i δ/ − V ′ ( x − w j − i δ/ (cid:19) (A19)with V in (A17). We define f ( z ) ≡ ∂ z [ ζ ( z ) − ℘ ( z )] andobserve that the generalizations of the second and fourthidentities in (A12) are ∂ z α ( z ) = V ′ ( z ) + f ( z ) (A20) and ∂ x (cid:2) α ( x − a ) α ( x − b ) (cid:3) = ∂ x (cid:2) α ( x − a ) − α ( x − b )] × α ( a − b ) + 12 (cid:2) f ( x − a ) + f ( x − b ) (cid:3) , (A21)respectively (the latter follows from the following well-known functional equation satisfied by the Weierstrassfunctions, [ ζ ( x ) + ζ ( y ) + ζ ( z )] = ℘ ( x ) + ℘ ( y ) + ℘ ( z )provided x + y + z = 0). The first and third identities in(A12) hold true as they stand.While f ( z ) = 0 in the hyperbolic case, it is a nontrivialfunction in the elliptic case. However, going through thecomputations described in Appendix A 1 c, one finds thatthey generalize straightforwardly to the elliptic case pro-vided M = N (that (A13) for M = N implies (A14) evenin the elliptic case has been known for a long time ).One thus obtains the same result as in the hyperboliccase but with the restriction M = N . Appendix B: Numerical method
We verified our soliton solutions numerically by adapt-ing a method developed for solving the standard ILWequation to the nonchiral ILW equation (6). The nu-merical method applies to the periodic problem on theinterval [ − L/ , L/ t , such that u ( x, t ) and v ( x, t ) are significantly differentfrom zero only in an interval [ − ℓ/ , ℓ/
2] with 0 < ℓ ≪ L ,this is an excellent approximation for the nonperiodicproblem on R . We thus checked numerically various 2-and 3-soliton solutions both for the periodic and nonpe-riodic problem, and we found excellent agreement. Forexample, the two-soliton solution in Fig. 3 computed withour numerical method cannot be distinguished with bareeyes from the one obtained with our analytic result. Wemention in passing that our numerical method is muchmore stable for initial conditions which give rise to soli-ton solutions than for generic initial conditions. In whatfollows, we describe our numeric method in more detail.We employ the discrete Fourier transform u ( x, t ) ≈ N − X n = − N ˆ u n ( t )e i k n x , k n ≡ n πL ˆ u n ( t ) = 12 N N − X j = − N u ( x j )e − i k n x j , x j ≡ j L N (B1)and the Fourier multiplier representations (A3) of thesingular integral operators (7), [ ( T u ) n ( t ) = i coth( k n δ )ˆ u n ( t ) , [ ( ˜ T u ) n ( t ) = i 1sinh( k n δ ) ˆ u n ( t ) , (B2)4to obtain a system of ordinary differential equations forthe time evolution of the Fourier coefficients via a semi-discrete collocation approximation (note that ˆ u n ( t ) /L can be identified with the Fourier transform ˆ u ( k n , t )).The numerical approximation for the nonlinear terms is \ (2 uu x ) n ( t ) = i k n d ( u ) n ( t ) with d ( u ) n ( t ) ≈ X m ˆ u n − m ( t )ˆ u m ( t ) ( − N ≤ n ≤ N −
1) (B3)where the sum on the right-hand side is over the integers m in the range − N ≤ m ≤ N − − N ≤ n − m ≤ N −
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